Related papers: Isoparametric foliation and Yau conjecture on the …
I'm sorry that there is a gap in this paper.
In this paper, we prove new pinching theorems for the first eigenvalue of the Laplacian on compact hypersurfaces of the Euclidean space. These pinching results are associated with the upper bound for the first eigenvalue in terms of higher…
Motivated by the celebrated Schoen-Yau-Gromov-Lawson surgery theory on metrics of positive scalar curvature, we construct a double manifold associated with a minimal isoparametric hypersurface in the unit sphere. The resulting double…
We refine some classical estimates in Seiberg-Witten theory, and discuss an application to the spectral geometry of three-manifolds. In particular, we show that on a rational homology three-sphere $Y$, for any Riemannian metric the first…
Consider a compact Riemannian manifold M of dimension n whose boundary \partial M is totally geodesic and is isometric to the standard sphere S^{n-1}. A natural conjecture of Min-Oo asserts that if the scalar curvature of M is at least…
In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one…
We consider the first eigenvalue of the magnetic Laplacian with zero magnetic field on simply connected compact surfaces and we establish isoperimetric inequalities and upper bounds in terms of a bound on the gaussian curvature. As a…
An integral inequality for the singular p-laplacian is established for 3/2<p<2. As consequence, lower bounds for the first eigenvalue of the p-laplacian are obtained for minimal submanifolds and prescribed scalar curvature submanifolds in…
Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st…
In this paper, we study the first eigenvalue of Jacobi operator on an $n$-dimensional non-totally umbilical compact hypersurface with constant mean curvature $H$ in the unit sphere $S^{n+1}(1)$. We give an optimal upper bound for the first…
Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that…
Chern's conjecture states that a closed minimal hypersurface in the euclidean sphere is isoparametric if it has constant scalar curvature. When the number $g$ of distinct principal curvatures is greater than three, few satisfactory results…
Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds…
In this article, we establish a geometric lower bound for the first positive eigenvalue $\lambda^{(1)}_{1}$ of the rough Laplacian acting on $1$-forms for closed $2n$-dimensional Riemannian manifolds with nonvanishing Euler characteristic.…
We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be…
In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet…
Associated with isoparametric foliations of unit spheres, there are two classes of minimal surfaces $-$ minimal isoparametric hypersurfaces and focal submanifolds. By virtue of their rich structures, we find new series of minimizing cones.…
Combining the intrinsic and extrinsic geometry, we generalize Einstein manifolds to Integral-Einstein (IE) submanifolds. A Takahashi-type theorem is established to characterize minimal hypersurfaces with constant scalar curvature (CSC) in…
We give an estimate of the first eigenvalue of the Laplace operator on a complete noncompact stable minimal hypersurface $M$ in a complete simply connected Riemannian manifold with pinched negative sectional curvature. In the same ambient…
In this paper, we give pinching Theorems for the first nonzero eigenvalue $\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $\epsilon>0$, there…